Calculus Examples

$lim_{a \to 2 b} - \frac{12 b^{2} + 3 a^{2}}{a - 2 b}$

Step 1

Set up the limit as a left-sided limit.

$lim_{a \to {(2 b)}^{-}} - \frac{12 b^{2} + 3 a^{2}}{a - 2 b}$

Step 2

Since the function $\frac{12 b^{2} + 3 a^{2}}{a - 2 b}$ approaches $\infty$ , the negative constant $- 1$ times the function approaches $- \infty$ .

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Step 2.1

Consider the limit with the constant multiple $- 1$ removed.

$lim_{a \to {(2 b)}^{-}} \frac{12 b^{2} + 3 a^{2}}{a - 2 b}$

Step 2.2

Since the numerator is negative and the denominator $a - 2 b$ approaches zero and is less than zero for $a$ near $2 b$ to the left, the function increases without bound.

$\infty$

Step 2.3

Since the function $\frac{12 b^{2} + 3 a^{2}}{a - 2 b}$ approaches $\infty$ , the negative constant $- 1$ times the function approaches $- \infty$ .

$- \infty$

Step 3

Set up the limit as a right-sided limit.

$lim_{a \to {(2 b)}^{+}} - \frac{12 b^{2} + 3 a^{2}}{a - 2 b}$

Step 4

Since the function $\frac{12 b^{2} + 3 a^{2}}{a - 2 b}$ approaches $- \infty$ , the negative constant $- 1$ times the function approaches $\infty$ .

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Step 4.1

Consider the limit with the constant multiple $- 1$ removed.

$lim_{a \to {(2 b)}^{+}} \frac{12 b^{2} + 3 a^{2}}{a - 2 b}$

Step 4.2

Since the numerator is negative and the denominator $a - 2 b$ approaches zero and is greater than zero for $a$ near $2 b$ to the right, the function decreases without bound.

$- \infty$

Step 4.3

Since the function $\frac{12 b^{2} + 3 a^{2}}{a - 2 b}$ approaches $- \infty$ , the negative constant $- 1$ times the function approaches $\infty$ .

$\infty$

Step 5

Since the left-sided limit is not equal to the right-sided limit, the limit does not exist.

$Does not exist$